## (1)

The table below shows

all the whole numbers

as the base numbers &

the respective triangular numbers

generated from these.

##### Triangular Numbers Table

##### Base

number

##### Triangular

number

•••••

•

••

•••

••••

•••••

0

(0 x 1)/2 = 0

1

(1 x2 )/2 = 1

4

(4 x 5)/2 = 10

9

(9 x 10)/2 = 45

16

(16 x 17)/2 = 136

•

•

•

•

•

•

•

•

## (2)

The tabulation below shows

all the sub-totals of

the numbers per (1).

0 = 0

0 + 1 = 1

0 + 1 + 10 = 11

0 + 1 + 10 + 45 = 56

0 + 1 + 19 + 45 + 136 = 192

•

•

•

•

## (3)

The tabulation below shows

the numbers per (2)

multiplied by 60

& the resulting products

increased by

respective uneven numbers

starting from 1.

60(0) + 1 = 1

60(1) + 3 = 63

60(11) + 5 = 665

60(56) + 7 = 3367

60(192) + 9 = 11529

•

•

•

•

## (4)

The tabulation below shows

all the sub-totals of

the numbers per (3),

resulting in all whole numbers

to the 6th. power.

1 = 1 = 16

1 + 63 = 64 = 26

1 + 63 + 665 = 729 = 36

1 + 63 + 665 + 3367 = 4096 = 46

1 + 63 + 665 + 3367 + 11529 = 15625 = 56

•

•

•

•

From the tabulation above

one can deduce that in general

n to the 6th. power is

the summation of triangular numbers

as shown below:

n4 =

n-1

Σ

k=0

n-1

[Σ

k=0

[30k2(k6 + 1) + (2k + 1)]

]